Charting the path of our geometric knowledge

March 11, 2007|By Nathan L. Harshman

The Poincare Conjecture: In Search of the Shape of the Universe

By Donal O'Shea

Walker, 293 pages, $26.95

In mathematics, truth is less important than the proof of that truth. And the standard of proof is far more punishing than legal requirements such as "preponderance of evidence" or "beyond a reasonable doubt." Whereas unprovability is the status quo in most fields, even the proof that something cannot be proved is a positive and possibly career-making result for a mathematician.

In 1904, Henri Poincare, French mathematician and physicist, asked a question about the topology of three-dimensional spaces. Almost immediately, experts agreed on the answer to the question, and they named its answer the Poincare conjecture. However, the conjecture's proof eluded scores of the world's best mathematicians for decades, and only emerged, with fanfare and controversy, 100 years later. "The Poincare Conjecture: In Search of the Shape of the Universe," by Donal O'Shea, places the original question, its answer and the all-important proof into a sweeping historical context.

Over the history of mathematical thought, the same cycle of discovery repeats many times. An important mathematical statement is assumed to be true, and then aspiring scholars hurl themselves at it for a generation (or generations). Sometimes a proof is discovered, bringing laurels to the person who finally cracked it. Other times, the statement is eventually shown to be unprovable. Or occasionally, conventional wisdom is wrong and the statement is proven false. O'Shea, professor of mathematics and dean of faculty at Mt. Holyoke College in Massachusetts, explains how whole new branches of mathematics have sprung from the search for a proof to a generally accepted truth.

One example of this dynamic is the case of Euclid's fifth postulate. That ancient Greek mathematician built the whole edifice of geometry on a foundation of only five postulates. Because of the power and simplicity of Euclid's work, it transcended its origins and became, according to O'Shea, a competitor with the Bible and the Koran for most-read and most-translated book of all time. The details of how it was traded and translated between languages and cultures for more than 2,000 years make for thought-provoking reading.

But Euclid's last postulate is the least simple of the five, and it has been a worrisome pebble in the shoe of mathematicians for a long time. It can be stated in many logically equivalent ways, one of which is, parallel lines exist and they never intersect. This statement appears true, but is it really necessary?

Starting in the Enlightenment, it seems every natural philosopher and amateur mathematician attempted to prove it from the other four postulates, without success. As mathematician Farkas Bolyai warned his son Janos:

"I implore you to make no attempt to master the theory of parallels. . . . Fear it no less than sensual passions, because it too may take all your time, deprive you of your health, peace of mind and happiness in life."

But O'Shea shows that, just like chasing "sensual passions," the single-minded, relentless pursuit of proof can be a creative process. The younger Bolyai ignored his father's advice and ended up one of three 19th Century mathematicians credited with inventing non-Euclidean geometry. They showed that the fifth postulate was not provable from the other four, but truly an axiom of one specific type of geometry, now called Euclidean geometry. Alternate versions of the fifth postulate lead to alternate types of geometry, and these new geometries are necessary to understand the space-time of the relativistic universe.

The Poincare conjecture is, like Euclid's fifth postulate, a statement about shapes and spaces. To get his audience to understand and appreciate the conjecture and its importance, O'Shea gives a grand tour of geometry and topology. He introduces concepts like higher dimensions, manifolds and spatial curvature in an intuitive and gradual manner. This topic is very visual, and diagrams instruct the reader in place of formulas and precise definitions, although some important ones are available in the extensive end notes for reference by the more mathematically experienced reader.

The central topological object of the Poincare conjecture is handled deftly by O'Shea: the shape called the three-sphere. As the circle is to the two-sphere (better known to non-mathematicians as the surface of a ball), the two-sphere is to the three-sphere. O'Shea has a large bag of tricks for inducing our typically three-dimensional imaginations to step into the fourth dimension.